The Renormalization Scale Setting Problem in QCD [Ph.D. Thesis]

Abstract

A key issue in making precise predictions in perturbative QCD is the uncertainty in setting the renormalization scale. If in principle, the entire perturbative series is void of this issue, in practice the perturbative corrections are known up to a certain order of accuracy and scale invariance is only approximated leading to the so-called scheme and scale ambiguities. According to the conventional scale setting (CSS) this problem cannot be avoided and the renormalization scale is set to the typical scale of a process $Q$, and errors are estimated by varying the scale over a range of two $[Q/2;2Q]$. This method is not void of ambiguities and leads to predictions affected by large theoretical errors. Other strategies for the optimization of the truncated expansion, such as the Principle of Minimal Sensitivity (PMS) and the Fastest Apparent Convergence (FAC) criterion, have the same difficulties of CSS and lead to incorrect and unphysical results. In general a scale-setting procedure is considered reliable if it preserves important self-consistency requirements such as all the renormalization group properties. Other requirements are suggested by known tested theories, by the convergence behavior of the series or by phenomenological results and scheme independence. All theoretical requirements for a reliable scale-setting procedure can be satisfied at once, leading to accurate results by using the Principle of Maximum Conformality (PMC). This method generalizes the Brodsky-Lepage-Mackenzie (BLM) method to all orders and to all observables and in the perspective of a theory unifying all interactions, such as the grand unified theory (GUT), the PMC offers the possibility to apply the same method in all sectors of a theory, starting from first principles, eliminating the renormalon growth, the scheme and scale ambiguities, and satisfying the QED Gell-Mann-Low scheme in the $N_c\to 0$ limit.